Dilogarithm of Minus One

Theorem

$\map {\Li_2} {-1} = -\dfrac 1 2 \map \zeta 2$

where:

$\map {\Li_2} x$ is the dilogarithm function of $x$

$\map \zeta 2$ is the Riemann $\zeta$ function of $2$.

Proof

\(\ds \map {\Li_2} z\)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}\)

Power Series Expansion for Spence's Function

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} {n^2}\)

$z := -1$

\(\ds \)

\(=\)

\(\ds - \map \eta 2\)

Definition of Dirichlet Eta Function

\(\ds \)

\(=\)

\(\ds - \paren {1 - 2^{1 - 2} } \map \zeta 2\)

Riemann Zeta Function in terms of Dirichlet Eta Function

\(\ds \)

\(=\)

\(\ds -\dfrac 1 2 \map \zeta 2\)

$\blacksquare$